## 17Calculus - The Unit Impulse Function and The Laplace Transform

Unit Impulse Function

The unit impulse function is kind of a strange function. You can see the graph on the right. Even though we use an arrow in the graph, this is not a vector. The arrow represents the fact that the function height is infinity. We draw it from zero to one to represent that the area is one. Like the unit step function, this function can be used to filter another function. The unit impulse function filters a function at a specific point.

Formally, the unit impulse is a mathematical construct whose properties are that the area under the graph is one, with infinite height and zero width. Sounds impossible, right? Well, not in mathematics (math is cool, eh?)!

Unit Impulse Function

$$\displaystyle{ \delta (t) = \left\{ \begin{array}{lr} \infty & t = 0 \\ 0 & t \neq 0 \end{array} \right. }$$

$$\displaystyle{ \int_{-\infty}^{+\infty}{\delta(t)~dt} = 1 }$$

Here is a video explaining this in more detail.

### Houston Math Prep - Unit Impulse & Dirac Delta Function [11min-33secs]

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